Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Properties of the Intervals of Real Numbers

**by****Jozef Bialas**- Received January 12, 1993
- MML identifier: MEASURE5

- [ Mizar article, MML identifier index ]

environ vocabulary SUPINF_1, ARYTM_3, RLVECT_1, ARYTM_1, ORDINAL2, RCOMP_1, BOOLE, MEASURE5, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XREAL_0, REAL_1, SUPINF_1, SUPINF_2; constructors REAL_1, SUPINF_2, MEMBERED, XBOOLE_0; clusters SUBSET_1, XREAL_0, XBOOLE_0, MEMBERED, ZFMISC_1; requirements NUMERALS, SUBSET, BOOLE, ARITHM; begin :: Some theorems about R_eal numbers reserve x,y,a,b,a1,b1,a2,b2 for R_eal; theorem :: MEASURE5:1 x <> -infty & x <> +infty & x <=' y implies 0. <=' y - x; theorem :: MEASURE5:2 (not (x = -infty & y = -infty) & not (x = +infty & y = +infty) & x <=' y) implies 0. <=' y - x; canceled 5; theorem :: MEASURE5:8 for a,b,c being R_eal holds (b <> -infty & b <> +infty & not (a = -infty & c = -infty) & not (a = +infty & c = +infty)) implies (c - b) + (b - a) = c - a; theorem :: MEASURE5:9 inf{a1,a2} <=' a1 & inf{a1,a2} <=' a2 & a1 <=' sup{a1,a2} & a2 <=' sup{a1,a2}; definition let a,b be R_eal; func [.a,b.] -> Subset of REAL means :: MEASURE5:def 1 for x being R_eal holds x in it iff a <=' x & x <=' b & x in REAL; func ].a,b.[ -> Subset of REAL means :: MEASURE5:def 2 for x being R_eal holds x in it iff (a <' x & x <' b & x in REAL); func ].a,b.] -> Subset of REAL means :: MEASURE5:def 3 for x being R_eal holds x in it iff (a <' x & x <=' b & x in REAL); func [.a,b.[ -> Subset of REAL means :: MEASURE5:def 4 for x being R_eal holds x in it iff (a <=' x & x <' b & x in REAL); end; definition let IT be Subset of REAL; attr IT is open_interval means :: MEASURE5:def 5 ex a,b being R_eal st a <=' b & IT = ].a,b.[; attr IT is closed_interval means :: MEASURE5:def 6 ex a,b being R_eal st a <=' b & IT = [.a,b.]; end; definition cluster open_interval Subset of REAL; cluster closed_interval Subset of REAL; end; definition let IT be Subset of REAL; attr IT is right_open_interval means :: MEASURE5:def 7 ex a,b being R_eal st a <=' b & IT = [.a,b.[; synonym IT is left_closed_interval; end; definition let IT be Subset of REAL; attr IT is left_open_interval means :: MEASURE5:def 8 ex a,b being R_eal st a <=' b & IT = ].a,b.]; synonym IT is right_closed_interval; end; definition cluster right_open_interval Subset of REAL; cluster left_open_interval Subset of REAL; end; definition let IT be Subset of REAL; attr IT is interval means :: MEASURE5:def 9 IT is open_interval or IT is closed_interval or IT is right_open_interval or IT is left_open_interval; end; definition cluster interval Subset of REAL; end; definition mode Interval is interval Subset of REAL; end; reserve A,B for Interval; definition cluster open_interval -> interval Subset of REAL; cluster closed_interval -> interval Subset of REAL; cluster right_open_interval -> interval Subset of REAL; cluster left_open_interval -> interval Subset of REAL; end; canceled; theorem :: MEASURE5:11 for x being set, a,b being R_eal st (x in ].a,b.[ or x in [.a,b.] or x in [.a,b.[ or x in ].a,b.]) holds x is R_eal; theorem :: MEASURE5:12 for a,b being R_eal st b <' a holds ].a,b.[ = {} & [.a,b.] = {} & [.a,b.[ = {} & ].a,b.] = {}; theorem :: MEASURE5:13 for a being R_eal holds ].a,a.[ = {} & [.a,a.[ = {} & ].a,a.] = {}; theorem :: MEASURE5:14 for a being R_eal holds ((a = -infty or a = +infty) implies [.a,a.] = {}) & ((a <> -infty & a <> +infty) implies [.a,a.] = {a}); theorem :: MEASURE5:15 for a,b being R_eal st b <=' a holds ].a,b.[ = {} & [.a,b.[ = {} & ].a,b.] = {} & [.a,b.] c= {a} & [.a,b.] c= {b}; theorem :: MEASURE5:16 for a,b,c being R_eal st a <' b & b <' c holds b in REAL; theorem :: MEASURE5:17 for a,b being R_eal st a <' b ex x being R_eal st a <' x & x <' b & x in REAL; theorem :: MEASURE5:18 for a,b,c being R_eal st a <' b & a <' c ex x being R_eal st a <' x & x <' b & x <' c & x in REAL; theorem :: MEASURE5:19 for a,b,c being R_eal st a <' c & b <' c ex x being R_eal st a <' x & b <' x & x <' c & x in REAL; theorem :: MEASURE5:20 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in ].a2,b2.[) or (not x in ].a1,b1.[ & x in ].a2,b2.[)); theorem :: MEASURE5:21 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in ].a2,b2.[) or (not x in ].a1,b1.[ & x in ].a2,b2.[)); theorem :: MEASURE5:22 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in ].a2,b2.[) or (not x in [.a1,b1.] & x in ].a2,b2.[)); theorem :: MEASURE5:23 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in ].a2,b2.[) or (not x in [.a1,b1.] & x in ].a2,b2.[)); theorem :: MEASURE5:24 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in [.a2,b2.]) or (not x in ].a1,b1.[ & x in [.a2,b2.])); theorem :: MEASURE5:25 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in [.a2,b2.]) or (not x in ].a1,b1.[ & x in [.a2,b2.])); theorem :: MEASURE5:26 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in [.a2,b2.[) or (not x in ].a1,b1.[ & x in [.a2,b2.[)); theorem :: MEASURE5:27 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in [.a2,b2.[) or (not x in ].a1,b1.[ & x in [.a2,b2.[)); theorem :: MEASURE5:28 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in ].a2,b2.[) or (not x in [.a1,b1.[ & x in ].a2,b2.[)); theorem :: MEASURE5:29 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in ].a2,b2.[) or (not x in [.a1,b1.[ & x in ].a2,b2.[)); theorem :: MEASURE5:30 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in ].a2,b2.]) or (not x in ].a1,b1.[ & x in ].a2,b2.])); theorem :: MEASURE5:31 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.[ & not x in ].a2,b2.]) or (not x in ].a1,b1.[ & x in ].a2,b2.])); theorem :: MEASURE5:32 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in ].a2,b2.[) or (not x in ].a1,b1.] & x in ].a2,b2.[)); theorem :: MEASURE5:33 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in ].a2,b2.[) or (not x in ].a1,b1.] & x in ].a2,b2.[)); theorem :: MEASURE5:34 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in [.a2,b2.]) or (not x in [.a1,b1.] & x in [.a2,b2.])); theorem :: MEASURE5:35 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in [.a2,b2.]) or (not x in [.a1,b1.] & x in [.a2,b2.])); theorem :: MEASURE5:36 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in [.a2,b2.[) or (not x in [.a1,b1.] & x in [.a2,b2.[)); theorem :: MEASURE5:37 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in [.a2,b2.[) or (not x in [.a1,b1.] & x in [.a2,b2.[)); theorem :: MEASURE5:38 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in [.a2,b2.]) or (not x in [.a1,b1.[ & x in [.a2,b2.])); theorem :: MEASURE5:39 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in [.a2,b2.]) or (not x in [.a1,b1.[ & x in [.a2,b2.])); theorem :: MEASURE5:40 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in ].a2,b2.]) or (not x in [.a1,b1.] & x in ].a2,b2.])); theorem :: MEASURE5:41 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.] & not x in ].a2,b2.]) or (not x in [.a1,b1.] & x in ].a2,b2.])); theorem :: MEASURE5:42 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in [.a2,b2.]) or (not x in ].a1,b1.] & x in [.a2,b2.])); theorem :: MEASURE5:43 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in [.a2,b2.]) or (not x in ].a1,b1.] & x in [.a2,b2.])); theorem :: MEASURE5:44 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in [.a2,b2.[) or (not x in [.a1,b1.[ & x in [.a2,b2.[)); theorem :: MEASURE5:45 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in [.a2,b2.[) or (not x in [.a1,b1.[ & x in [.a2,b2.[)); theorem :: MEASURE5:46 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in ].a2,b2.]) or (not x in [.a1,b1.[ & x in ].a2,b2.])); theorem :: MEASURE5:47 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in [.a1,b1.[ & not x in ].a2,b2.]) or (not x in [.a1,b1.[ & x in ].a2,b2.])); theorem :: MEASURE5:48 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in [.a2,b2.[) or (not x in ].a1,b1.] & x in [.a2,b2.[)); theorem :: MEASURE5:49 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in [.a2,b2.[) or (not x in ].a1,b1.] & x in [.a2,b2.[)); theorem :: MEASURE5:50 a1 <' a2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in ].a2,b2.]) or (not x in ].a1,b1.] & x in ].a2,b2.])); theorem :: MEASURE5:51 b1 <' b2 & (a1 <' b1 or a2 <' b2) implies ex x being R_eal st ((x in ].a1,b1.] & not x in ].a2,b2.]) or (not x in ].a1,b1.] & x in ].a2,b2.])); theorem :: MEASURE5:52 (a1 <' b1 & ((A = ].a1,b1.[ or A = [.a1,b1.] or A = [.a1,b1.[ or A = ].a1,b1.]) & (A = ].a2,b2.[ or A = [.a2,b2.] or A = [.a2,b2.[ or A = ].a2,b2.]))) implies (a1 = a2 & b1 = b2); definition let A be Interval; func vol A -> R_eal means :: MEASURE5:def 10 ex a,b being R_eal st ((A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.]) & (a <' b implies it = b - a) & (b <=' a implies it = 0.)); end; theorem :: MEASURE5:53 for A being open_interval Subset of REAL holds for a,b being R_eal holds A = ].a,b.[ implies ((a <' b implies vol(A) = b - a) & (b <=' a implies vol(A) = 0.)); theorem :: MEASURE5:54 for A being closed_interval Subset of REAL holds for a,b being R_eal holds A = [.a,b.] implies ((a <' b implies vol(A) = b - a) & (b <=' a implies vol(A) = 0.)); theorem :: MEASURE5:55 for A being right_open_interval Subset of REAL holds for a,b being R_eal holds A = [.a,b.[ implies ((a <' b implies vol(A) = b - a) & (b <=' a implies vol(A) = 0.)); theorem :: MEASURE5:56 for A being left_open_interval Subset of REAL holds for a,b being R_eal holds A = ].a,b.] implies ((a <' b implies vol(A) = b - a) & (b <=' a implies vol(A) = 0.)); theorem :: MEASURE5:57 for a,b,c being R_eal holds (a = -infty & b in REAL & c = +infty & (A = ].a,b.[ or A = ].b,c.[ or A = [.a,b.] or A = [.b,c.] or A = [.a,b.[ or A = [.b,c.[ or A = ].a,b.] or A = ].b,c.])) implies vol(A) = +infty; theorem :: MEASURE5:58 for a,b being R_eal holds (a = -infty & b = +infty & (A = ].a,b.[ or A = [.a,b.] or A = [.a,b.[ or A = ].a,b.])) implies vol(A) = +infty; definition cluster empty Interval; end; definition redefine func {} -> empty Interval; end; canceled; theorem :: MEASURE5:60 vol {} = 0.; theorem :: MEASURE5:61 (A c= B & B =[.a,b.] & b <=' a) implies (vol(A) = 0. & vol(B) = 0.); theorem :: MEASURE5:62 A c= B implies vol A <=' vol B; theorem :: MEASURE5:63 0. <=' vol(A);

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